| L(s) = 1 | + 0.461·2-s − 3·3-s − 7.78·4-s − 5·5-s − 1.38·6-s − 7.28·8-s + 9·9-s − 2.30·10-s − 24.3·11-s + 23.3·12-s + 66.1·13-s + 15·15-s + 58.9·16-s + 19.6·17-s + 4.15·18-s − 64.6·19-s + 38.9·20-s − 11.2·22-s + 147.·23-s + 21.8·24-s + 25·25-s + 30.5·26-s − 27·27-s + 166.·29-s + 6.91·30-s − 189.·31-s + 85.4·32-s + ⋯ |
| L(s) = 1 | + 0.163·2-s − 0.577·3-s − 0.973·4-s − 0.447·5-s − 0.0941·6-s − 0.321·8-s + 0.333·9-s − 0.0729·10-s − 0.666·11-s + 0.561·12-s + 1.41·13-s + 0.258·15-s + 0.920·16-s + 0.279·17-s + 0.0543·18-s − 0.781·19-s + 0.435·20-s − 0.108·22-s + 1.33·23-s + 0.185·24-s + 0.200·25-s + 0.230·26-s − 0.192·27-s + 1.06·29-s + 0.0421·30-s − 1.10·31-s + 0.471·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.461T + 8T^{2} \) |
| 11 | \( 1 + 24.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 66.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 19.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 64.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 147.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 166.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 189.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 75.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 186.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 110.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 380.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 712.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 308.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 489.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 757.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 58.0T + 3.57e5T^{2} \) |
| 73 | \( 1 + 58.0T + 3.89e5T^{2} \) |
| 79 | \( 1 - 865.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 829.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.09e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.11e3T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.500709762893109450482955716349, −8.660054668922995490857425618740, −7.993403985277030441077551164214, −6.82138918477865835640504464448, −5.81861830549520750482368630091, −4.99561490269359742148421429031, −4.10924827789934818087706311333, −3.14138153783314693025219269258, −1.19589978091034030138149270723, 0,
1.19589978091034030138149270723, 3.14138153783314693025219269258, 4.10924827789934818087706311333, 4.99561490269359742148421429031, 5.81861830549520750482368630091, 6.82138918477865835640504464448, 7.993403985277030441077551164214, 8.660054668922995490857425618740, 9.500709762893109450482955716349
